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Thank you very much for the kind invitation and also for your introduction.

I think nobody has said so much about the afternoon.

It is really a pleasure to be here in Erlangen.

It's not my first visit, but they are always very fruitful, I would say, but also very pleasant.

And this is a special opportunity which really honors me that I'm allowed, I mean, you thought about me to give a talk here.

Oops, no, I'm lost. Not this one, this one.

So, yeah, when you invited me, I thought about what should I talk about?

And, well, I thought there should not be much interference with what Hans van Dijn was going to talk,

so I thought, well, that's a good subject to speak about things on flowing porous media numerical methods

and fixed mixed finite elements, so that's actually a good subject to mention here,

although I shouldn't go into many details here because I'm sure there are more experts in the room,

so, yeah, I could easily run into troubles.

So, as mentioned, I mean, this is my talk is on numerical methods for reactive flow in porous media,

and, well, I didn't mention all because, of course, it is not a work which I did by myself,

it's also involving a lot of people from whom I learned.

Well, just to be very brief about why do we do this, why we are interested in reactive flow in porous media,

of course, there are many applications. If you look at only this picture, basically, this is basically flow through soils,

and we see we have plants there, and plants have to take nutrients, so this means there is water that has to transport nutrients there,

and nutrients are usually dissolved or adsorbed in the soil and then brought to the plants and so on.

Well, it's also related to contamination of various sites in the subsurface. Again, you transport, there are contaminants in the soil,

and then this can go even into phreatic water, so basically what you want to do is to remediate this problem by injecting other substances

that can neutralize the effect of contaminants. More in the past years, there was more interest in geological CO2 sequestration,

so basically we realized that we produced in accelerated tempo CO2 in the atmosphere, which is pretty bad for our environment,

so one option is to capture it at the production sites, for example at various plants, and to store it into the subsurface.

Well, these are all examples involving actually flow in the subsurface, so in particular water, but not only, it can be actually supercritical,

CO2 or oil or whatever, but it's not only the flow, but it's also the advection and dispersion of multiple components,

and basically there are various kinds, I mean we saw in the talk of Hans van Dijn yesterday, there are various ways to model these kind of problems,

including equilibrium and non-equilibrium adsorption, I mean adsorption or desorption, and to me the most, I mean a very challenging issue is

if you couple actually this kind of variable saturated flow to reactive flows. So I'm going to go into details a few slides later,

but let me mention first of all that okay, there are various issues related to this kind of reactive flow in porous media.

First of all is more analysis, basically you want to know if the mathematical models are well posed, do they have a solution in which sense,

how many solutions, all this stuff, but also basically if you also for societal relevance, you're not going to come to an engineer

and tell the engineers okay, there is a solution in whichever abstract space because people usually have not much out of these kind of

answers, but you also show pictures and numbers, and for this you need numerical schemes and to show that they are efficient,

so they converge, they have certain accuracy and so on. And in particular one question we were triggered during this kind of research

was well, if the accuracy in computing the flow of the two fluids or however, how many you have, one in the subsurface

will influence the accuracy in the reaction. So basically the goal is to speak about reactive flow and reactions in porous media.

So basically if this is the question, then if it's a flow in porous media and you start studying actually these kind of problems,

I'm sure you easily, you immediately come across Erlangen. It's not that all flows are through Erlangen, but the research on mathematical

aspects on such kind of problems, they are so strong in Erlangen that you really have to consider this to study what's being done

in the team coached by Peter Knabner. So basically for me it was a great opportunity to meet you actually in Algorithmy,

I think it was 97, very long time ago, when you mentioned that you asked me about Romanian students you were planning to bring to Erlangen.

Well there was a follow-up three years later at a conference in Heidelberg honoring Vili Jager, and there you told me that you brought

the students to Erlangen, and one of the students was working on a problem I was also working on, I mean on Richard's equation.

So this was actually for me a very fortunate encounter because the student you were mentioning was one of the strikers you had on the slides

before, and that was Florin Radu, and this was actually he came to Eindhoven and this was the starting point of a long collaboration which gave

actually us several papers, and we have a couple of them in journals but also in some proceedings, and they were mainly on degenerate

parabolic problems, modeling either unsaturated or two-phase flow in porous media. Of course this is not only joint publication which I want to

mention but actually there is the aspect on reactive transport and in particular dissolution and precipitation, and this at least for me